Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}\) and \(\vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}\), then a unit vector perpendicular to both vectors \((\vec{a}+\vec{b})\) and \((\vec{a}-\vec{b})\) is equal to

If \(e^y(x+1)=1\) and \(\frac{d^2 y}{d x^2}=k\left(\frac{d y}{d x}\right)^2\), then \(k\) is equal to

The simplified form of \(\tan ^{-1}\left(\frac{\cos x}{1+\sin x}\right),-\frac{\pi}{2}< x <\frac{\pi}{2}\) is

The number of equivalence relation on the set {1, 2, 3} containing (1, 2) and (2, 1) is

The probability distribution of a random variable X is given by

X	0	1	2
P(X)	\(1-7 a^2\)	\(\frac{1}{2} a+\frac{1}{4}\)	\(a^2\)

If \(a>0\), then \(P(0<X \leq 2)\) is equal to

The value of \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(x^5+x^3+x+2\right) d x\) is:

The largest interval, in which the function \(f(x)=x^3+2 x^2-1\) is increasing, is:

Observe the given figure and answer the question.


Which one of the following is not shown in the above figures?

A thin convex lens of refractive index 1.5 has focal length f1. When immersed in a liquid of refractive index 5/3, its focal length is found to be f2. The relationship between f1 and f2 is:

In reference to Bt cotton, select the correct statement(s) :
(A) The alkaline pH of an insect gut solublises the crystals of Bt toxin
(B) The acidic pH of an insect gut solublises the crystals of Bt toxin
(C) The Bt toxin does not kill the tobacco budworm
(D) Specific Bt toxin genes were isolated from Bacillus thuringiensis
Choose the correct answer from the options given below :

Read the passage carefully and answer the questions
Carboxylic acids are the organic compounds which comprise of carboxyl functional group, -COOH. These are generally found in nature, some higher members are known as fatty acids as they occur in natural fats as esters of glycerol. They can be prepared from oxidation of alcohols and aldehydes, hydrolysis of nitriles and amides and Grignard reagents. They are stronger acids than simple phenols and alcohols. but weaker than mineral acids. Substituents affect the acidity of aliphatic and aromatic carboxylic acids. They undergo reactions involving cleavage of -OH and C-OH bond. Carboxylic acids having an alpha hydrogen get halogenated in the presence of red phosphorus to form alpha halo carboxylic acids. The reaction is known as Hell Volhard Zelinsky reaction. Besides these aromatic carboxylic acids undergo electrophilic substitution reactions. They are versatile and are employed in the manufacture of various useful compounds. For example, methanoic acid is used in rubber, textile, dyeing, leather and electroplating industries. Higher fatty acids are used in the manufacture of soaps and detergents. Esters of Benzoic acid are used in perfumery.
Which carboxylic acid will not undergo HVZ reaction?

Read the following passage carefully and answer the given questions.
Bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus Trichoderma polysporum. Statins produced by the yeast Monascus purpureus have been commercialised as blood-cholesterol lowering agents. It acts by competitively inhibiting the enzyme responsible for synthesis of cholesterol. Baculoviruses are pathogens that attack insects and other arthropods. The majority of baculoviruses used as biological control agents are in the genus Nucleopolyhedrovirus. These viruses are excellent candidates for species-specific, narrow spectrum insecticidal applications. They have been shown to have no negative impacts on plants, mammals, birds, fish or even on non-target insects. BOD refers to the amount of the oxygen that would be consumed if all the organic matter in one liter of water were oxidised by bacteria. The sewage water is treated till the BOD is reduced. Cheese, is one of the oldest food items in which microbes were used. Different varieties of cheese are known by their characteristic texture, flavour and taste. For example, the large holes in ‘Swiss cheese’ are due to production of a large amount of CO₂ by a bacterium named Propionibacterium sharmanii.
The greater the BOD of waste water-

If the matrix \(\left[\begin{array}{ccc}3 & 2 a & -5 \\ 4 & 0 & b \\ -5 & 3 & 10\end{array}\right]\) is symmetric, then the value of \(5 a+2 b\) is :